### Laser physics EAL 501

```Laser physics EAL 501
Lecture 3
Energy units
• 1 eV= 1.6x10-19 (C) x 1 V= 1.6x10-19 J
• E =hc/λ
• 1/λ=E/hc=1J/(6.6x10-34x108x100)
1 cm-1
=1.5 x10-23 J
=.00012 eV
Absorption, Emission, and Dispersion of Light
• Electron Oscillator Model
• Spontaneous Emission
• Absorption
• Thermal Radiation
• Emission and Absorption of Narrowband Light
• Collision Broadening
• Doppler Broadening
• The Voigt Profile
• Radiative Broadening
• Absorption and Gain Coefficients
Absorption of gases
• White light propagating through a gas is absorbed at
the resonance frequencies of the atoms or molecules.
Sodium, for instance, has strong absorption lines in the
yellow region at 589.0 and 589.6 nm
The absorbed energy is dissipated in
• Heat (translational kinetic energy of the atoms)
• Collision
• Resonance fluorescence :Re-emission in all directions (
• Radiation quenching :When the pressure of the gas is
increased, collisions may rapidly convert the absorbed
radiation into heat before it can be reradiated.
Lorentz model
• This hypothesis states that an electron in an atom
responds to light as if it were bound to its atom
or molecule by a simple spring. As a consequence
the electron can be imagined to oscillate about
the nucleus.
F x   kx
Fv    .v
2
F  m.
d x
dt
2
e . E  kx  
dx
2
 m
dt
2
d x
dt
2

 dx
m dt

d x
dt
k
m
2
x  e . E cos(  t )
Fe  eE ( x , t )
Electron Natural oscillation:
Solution if no external field and no friction
2
d x
2

k
x  x  A cos(  o t )   o 
k
dt
m
m
This corresponds to the spontaneous emission
In the presence of External field , the electron can absorb energy only if ω= ωo
The friction correspond to radiation losses in the material.
Semi classical view
spontaneous emission
Boltzman Law
N 2 / N1  e

E2 , N2
E 2  E1
k BT
E1 , N1
d
dt
N1  
d
dt
N 2  A 21 N 2
A21 is the rate of spontaneous emission = 1/ τ21 the level 2 life time
Or Decay time of level 2 to level 1
N 2 (t )  N 2 ( 0 ) e
For multi level decay
An 
A
m

t
 21
nm
Some forms of spontaneous emission
• Electroluminescence : If excitation occurs in an
electric discharge such as a spark.
• Chemiluminescence : If excitation produced as a
by-product of a chemical reaction.
• Bioluminescence : If excitation occurs in a living
organism (such as a firefly),.
• Fluorescence refers to spontaneous emission
from an excited state produced by the absorption
of light.
• Phosphorescence describes the situation in which
the emission persists long after the exciting light
is turned off and is associated with a metastable
Absorption
E2 , N2
d
dt
N 1   B12 N 1 I ( ) S ( )
S is the line shape
The simplest is the Lorentzian line shape L
L ( ) 
 o / 
( v   o )   o
2
2
E1 , N1
Spectral energy density of radiation
It is convenient to define a spectral energy density
ρ(ν), such that ρ(ν) d ν is the electromagnetic
energy per unit volume in the frequency band ν ,
ν+dν
The intensity, or energy flux, is the velocity of light
times the energy density. Therefore
I(ν)dν = cρ(ν)
Stimulated emission
E2 , N2
E1 , N1
d
dt
N 2   B 21 N 2 I ( )
Thermal Radiation
• A black body is a body that absorbs all the
energy incident
8 h  / c
3
 ( ) 
h
e
 max 
3
k BT
2898
T
1
[ m ]
Einstein relations
g 2 B12  g 1 B 21
B 21 
c
3
8 g 2 h 
3
A 21
So for a two level system in the presence of radiation we can write
d
dt
N2  
d
dt
N 1  B12  ( ) S ( ) N 1  B 21  ( ) S ( ) N 2  A 21 N 1
 B12  ( ) S ( )( N 1 
g1
g2
N 2 )  A 21 N 1
Line shapes
Natural line brodening : due to spontaneous emission
Lorentzian line shape
 rad 

A nm / 4 
Homogeneous line broadening : due to atomic collisions
broadening increase with pressure
 o  collisionr ate / 2
Inhomogeneous line broadening : due to
Doppler effect
Gaussian line shape
Propagation of light through a 2 level medium
Suppose a light of intensity Iv(0) is incident on the material.
The intensity In is equal to the energy density per unit volume times
the wave propagation velocity.
The rate at which electromagnetic energy passes through a plane
cross-sectional area A at z is Iv(z)A, and at an adjacent plane at z+dz
this rate is Iv(z+dz)A; the difference is

( I  ( z   z )  I  ( z )) A 
( I v ( z ) A )  dz
z
The rate at which energy is accumulated or depleted in the volume Δz is

t
( u ( ) A  z ) 

z
( I ( z ) A )  z
The change in radiation in the medium could be due to absorption
and emission. If we neglect the spontaneous emission for the
time being
The rate of increase of N2 equals the rate of decrease of the field so
(
(
(
1 
c t
1 
c t
1 
c t




z

z

z
) I   h  B 21 u ( ) S ( )( N 2 
) I 
h  B 21 S ( )
c
) I   g ( ) I  .
(N 2 
g2
g1
g2
g1
N1)
N 1 ) I .
The gain coefficient has dimension m-1 . If g >0 it is
amplification. If g<0 it is absorption
 A21
2
g ( ) 
8
S ( )( N 2 
g2
g1
N1)
Remember that the wave length here λ is the wave length inside
the material which equals the wavelength in vacuum divided by
the index of refraction of the material (the frequency doesn’t
change inside the material
 A21
2
g ( ) 
8 n
2
S ( )( N 2 
g2
g1
N1)
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